Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2008, Article ID 249438, 8 pages doi:10.1155/2008/249438 Research Article On Multivariate Gr ¨ uss Inequalities Chang-Jian Zhao 1 and Wing-Sum Cheung 2 1 Department of Information and Mathematics Sciences, College of Science, China Jiliang University, Hangzhou 310018, China 2 Department of Mathematics, The University of Hong Kong, Pokfulam Road, Hong Kong Correspondence should be addressed to Chang-Jian Zhao, chjzhao@163.com Received 6 March 2008; Revised 7 May 2008; Accepted 20 May 2008 Recommended by Martin Bohner The main purpose of the present paper is to establish some new Gr ¨ uss integral inequalities in n independent variables. Our results in special cases yield some of the recent results on Pachpatte’s, Mitrinovi´ c’s, and Ostrowski’s inequalities, and provide new estimates on such types of inequalities. Copyright q 2008 C.-J. Zhao and W.-S. Cheung. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction The well-known Gr ¨ uss integral inequality 1 can be stated as follows see 2, page 296: 1 b − a b a f xg xdx − 1 b − a b a f xdx 1 b − a b a g xdx ≤ 1 4 P − pQ − q, 1.1 provided that f and g are two integrable functions on a, b such that p ≤ f x ≤ P , q ≤ g x ≤ Q, for all x ∈ a, b, where p, P, q, Q are real constants. Many generalizations, extensions, and variants of this inequality 1.1 have appeared in the literature, see 1–8 and the references given therein. The main purpose of the present paper is to establish several multivariate Gr ¨ uss integral inequalities. Our results provide a new estimates on such type of inequalities. 2. Main results In what follows, R denotes the set of real numbers, R n the n-dimensional Euclidean space. Let D {x 1 ,...,x n : a i ≤ x i ≤ b i i 1,...,n}. For a function ux : R n → R, we denote the